where \(S\)  is van Neumann entropy and \(\hat{\rho }_{diag}\)  is a diagonal matrix which removing all off-diagonal elements of the density matrix \(\hat{\rho }\) .
In the case of Gaussian quantum information, a Gaussian state \(\hat{\rho }(\bar{\mathbf {x}},\mathbf {V})\)  can be completely represented by the displacement \(\bar{\mathbf {x}}\)  and the covariance matrix \(\mathbf {V}\)  in phase space, which correspond to the first and second statistical moments of the quadrature operators respectively [1]}, [2]}. The displacement \(\mathbf {\bar{x}}=\langle \mathbf {\hat{x}}\rangle \) , where \(\mathbf {\hat{x}=(}\mathit {\hat{X}}_{\mathit {1}}\mathit {,\mathit {\hat{Y}}_{\mathit {1}},...,\mathit {\hat{X}}}_{\mathit {\mathit {N}}}\mathit {,\hat{Y}}_{\mathit {N}}\mathbf {)}^{\mathit {t}},\mathit {\hat{X}}_{\mathit {k}}=(\hat{a}_{_{k}}+\hat{a}_{_{k}}^{\dagger })\)  and \(\mathit {\hat{Y}}_{\mathit {k}}=i(\hat{a}_{_{k}}^{\dagger }-\hat{a}_{_{k}})\)  are the amplitude and phase quadratures of an optical mode, respectively.
The element of covariance matrix \(\mathbf {V}\)  is defined as
\(\mathbf {V}_{ij}=\frac{1}{2}\langle \hat{x}_{i}\hat{x}_{j}+\hat{x}_{j}\hat{x}_{i}\rangle -\langle \hat{x}_{i}\rangle \langle \hat{x}_{j}\rangle \) . The diagonal Gaussian states (incoherent states) are thermal states [3]}, so the incoherent state \(\hat{\rho }_{diag}\)  in Eq. (1) is replaced by an \(N\) -mode thermal state \(\hat{\rho }(\bar{\mathbf {x}}_{th}, \mathbf {V}_{th})\)  whose mean number of particle is the same to \(\hat{\rho }(\bar{\mathbf {x}},\mathbf {V})\)  for each mode. Thus, the Gaussian quantum coherence of an \(N\) -mode Gaussian state can be represented as [3]}
\(\mathcal {C}_{rel.~ent.}\left[\hat{\rho }(\bar{\mathbf {x}},\mathbf {V})\right] =S\left[\hat{\rho }(\bar{\mathbf {x}}_{th},\mathbf {V}_{th})\right] -S\left[\hat{\rho }(\bar{\mathbf {x}},\mathbf {V}) \right],\) 
